Aptitude — Quick Summary

Quick revision: every topic, key terms, and mnemonics for Aptitude.

This is a quick revision doc covering all 48 topics in the Aptitude collection. Open the linked notes if you want depth — this is for re-cementing formulas, tricks, and common traps before an exam.

Foundations & Number Sense

Number Systems and Divisibility

What it is. Types of numbers (N, W, Z, Q, R) and rules to test divisibility by 2-12 fast.

Key formulas.

• Number of factors of N = (p+1)(q+1)(r+1) where N = aᵖ × bᑫ × cʳ
• Sum of factors = ∏ [(prime^(power+1) − 1) / (prime − 1)]
• Even factors = total − odd factors

Divisibility rules.

Worked. 360 = 2³×3²×5¹ → factors = 4×3×2 = 24.

Remember. 1 is neither prime nor composite. 2 is the only even prime. Memorize primes ≤ 50 (15 of them).

HCF and LCM

What it is. HCF = biggest divisor that fits all. LCM = smallest multiple all fit into.

Core formula. HCF × LCM = a × b (only for two numbers).

Methods.

• HCF prime factorization: lowest power of common primes
• HCF Euclid: divide larger by smaller, then divisor by remainder, until 0
• LCM prime factorization: highest power of all primes

Special.

• HCF of fractions = HCF(numerators) / LCM(denominators)
• LCM of fractions = LCM(numerators) / HCF(denominators)
• Largest number dividing X, Y leaving remainders r₁, r₂ → HCF(X−r₁, Y−r₂)
• Same remainder dividing X, Y, Z → HCF of pairwise differences

Trick. Bells/lights ringing together → LCM. Largest tile for rectangle → HCF.

Remember. HCF ≤ smaller ≤ larger ≤ LCM. Consecutive integers always coprime (HCF=1).

Fractions, Decimals, and Surds

What it is. Convert between forms quickly; rationalize denominators.

Fraction-decimal table (memorize): 1/2=.5, 1/3=.333, 1/4=.25, 1/5=.2, 1/6=.1667, 1/7=.1428, 1/8=.125, 1/9=.111, 1/10=.1, 1/11=.0909, 1/12=.0833, 1/16=.0625, 1/20=.05.

Recurring decimals.

• Pure recurring: 0.abc̄ = abc/999
• Mixed: 0.a b̄c̄ = (abc − a) / 990 (9s for repeating, 0s for non-repeating)

Surds.

• √2≈1.414, √3≈1.732, √5≈2.236, √7≈2.646
• Like surds add: 3√2 + 5√2 = 8√2
• Rationalize 1/(a+√b) by multiplying by conjugate (a−√b)

Trick. Compare √a − √b vs √c − √d: rationalize → 1/(√a+√b) etc., then compare denominators.

Remember. For consecutive numbers, gap between square roots shrinks as numbers grow.

Powers, Indices, and Roots

What it is. Laws of exponents and shortcuts for squares, cubes, roots.

Laws. aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ÷aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰=1; a⁻ⁿ=1/aⁿ; a^(m/n) = ⁿ√(aᵐ).

Memorize. Squares 1²–30², cubes 1³–15³. Powers of 2 (up to 1024) and 3 (up to 6561).

Squaring tricks.

• n5² = n(n+1) followed by 25. (35² = 12|25 = 1225)
• (50+x)² = (25+x)|x² (52² = 27|04 = 2704)
• (100+x)² = (100+2x)|x² (103² = 106|09 = 10609)

Compare powers. 2³⁰⁰ vs 3²⁰⁰ → reduce to common power: 2³ vs 3² → 8 vs 9 → 3²⁰⁰ wins.

Remember. Cube roots: unit digit of cube uniquely determines unit digit of root. 8↔2, 2↔8, others map to themselves.

Unit Digits and Remainders

What it is. Unit digits cycle; remainders follow patterns.

Cyclicity (period in parens).

• 0,1,5,6 → same forever (1)
• 4,9 → cycle of 2 (4→4,6,4,6 / 9→9,1,9,1)
• 2,3,7,8 → cycle of 4 (2→2,4,8,6 / 3→3,9,7,1 / 7→7,9,3,1 / 8→8,4,2,6)

Method for unit digit of aⁿ. Take unit digit of a, find cycle length, divide n by length, use remainder (remainder 0 → last in cycle).

Remainder shortcuts.

• ÷ 3 or 9: digit sum mod
• ÷ 11: alternating sum
• Fermat’s Little: if p prime and p∤a, then aᵖ⁻¹ ≡ 1 (mod p)

Remember. ALWAYS reduce base mod divisor first — huge speed-up. Sum of factorials ≥5! ends in 0, so unit digit only depends on 1!+2!+3!+4! = 33 → unit 3.

Simplification and Approximation

What it is. BODMAS + smart rounding for fast answers.

BODMAS. Brackets → Orders → Division/Multiplication (L→R) → Addition/Subtraction (L→R).

Tricks.

• 786 × 764 = (775+11)(775−11) = 775² − 121 (difference of squares)
• 19×21 = 20² − 1 = 399
• “Of” = ×
• Cancel common factors before multiplying chains

Approximation strategy. Round to nearest convenient number; round in opposite directions to cancel errors. 498.7×31.2/9.87 ≈ 500×31/10 = 1550.

Digit sum (cast out 9s). Verify products. Match digit sums of operands’ product with answer’s digit sum.

Remember. Use fraction-percentage table — 37.5% of 4816 = 3/8 of 4816 = 1806 (instant).

Arithmetic & Commercial Math

Percentages

What it is. Per hundred. Master fraction equivalents.

Fraction-percent table (must memorize): 12.5%=1/8, 16.67%=1/6, 20%=1/5, 25%=1/4, 33.33%=1/3, 37.5%=3/8, 40%=2/5, 50%=1/2, 62.5%=5/8, 66.67%=2/3, 75%=3/4, 87.5%=7/8.

Formulas.

• % = (Part/Whole) × 100
• % change = ((New−Old)/Old) × 100
• Successive change of a% and b%: a + b + ab/100
• Same x% up then x% down → net loss x²/100%
• Price up r%, keep expenditure same → reduce consumption by r/(100+r) × 100%

Reverse trick. After 20% increase value=600 → original = 600 × 5/6 = 500.

Remember. “A is x% more than B” ≠ “B is x% less than A.” Bases differ. 20% more ↔ 16.67% less. Population (1±r/100)ⁿ.

Profit, Loss, and Discount

What it is. CP → MP → SP. Markup adds; discount subtracts.

Formulas.

• Profit% = (Profit/CP) × 100; Loss% = (Loss/CP) × 100
• Profit/Loss% always on CP; Discount% always on MP
• Markup m% then discount d% → Profit = m − d − md/100
• Same SP for two articles, one at +x%, one at −x% → net loss x²/100
• “Buy X get Y free” → discount = Y/(X+Y) × 100%

Dishonest dealer. Uses w grams instead of 1000, sells at CP → Profit% = (1000−w)/w × 100.

Worked. Mark up 40%, discount 25% → 40 − 25 − 10 = 5% profit.

Remember. CP = SP × 100/(100±%). New SP = Old SP × (100+gain)/(100−loss).

Simple and Compound Interest

What it is. SI = simple, on principal only. CI = on principal + accumulated interest.

Formulas.

• SI = PRT/100; A = P(1 + RT/100)
• CI: A = P(1 + R/100)ⁿ; CI = A − P
• Half-yearly: rate R/2, periods 2n. Quarterly: R/4, 4n.

CI − SI shortcut (the most-tested trick).

• 2 years: CI − SI = P(R/100)²
• 3 years: P(R/100)² × (3 + R/100)

Doubling under SI. R × T = 100. If doubles in 5 yrs at SI, triples in 10, quadruples in 15.

Effective rate. R compounded n times/yr → (1 + R/n)ⁿ − 1.

Remember. Difference between successive years’ amounts = interest on previous year’s amount → directly gives the rate.

Ratio and Proportion

What it is. Comparison of quantities; manipulate via componendo-dividendo.

Formulas.

• a:b = ka:kb
• Mean proportional of a, c → b = √(ac)
• Componendo-dividendo: if a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d)
• Compounding: (a:b) × (c:d) = ac:bd

Combining. A:B = 2:3, B:C = 4:5 → make B common via LCM → A:B:C = 8:12:15.

Income−Expenditure. Set up two-equation system using Income − Expenditure = Savings.

Remember. Cross-multiply to compare fractions. Mean proportional uses geometric mean.

Averages

What it is. Sum / Count. Most useful trick: think in sums.

Formulas.

• Average = Sum/Count → Sum = Count × Avg
• Weighted: (n₁A₁ + n₂A₂)/(n₁+n₂)
• Equal distances at speeds a, b: avg speed = 2ab/(a+b) (harmonic, NOT a+b/2!)
• Equal times: (a+b)/2
• Three speeds equal distance: 3abc/(ab+bc+ca)

Tricks.

• Replace one element: change in avg = (new − old)/n
• Avg of n consecutive numbers = middle number
• Avg of first n natural numbers = (n+1)/2
• Avg of first n odd = n; first n even = n+1

Remember. Average speed for round trip = HARMONIC mean (2ab/(a+b)). Never (a+b)/2 unless equal time.

Mixtures and Alligation

What it is. Find mixing ratio for desired mean.

Alligation (cross) method.

• Cheaper:Dearer = (Dearer − Mean) : (Mean − Cheaper)

Works for ANY weighted-average problem (price, concentration, profit%, marks, age).

Repeated dilution. Remove x liters from V, replace with water, n times → original substance left = V(1 − x/V)ⁿ.

Worked. Tea Rs 40 + Rs 60 → mean Rs 45. Ratio = (60−45):(45−40) = 15:5 = 3:1.

Remember. Cross differences swap (cheaper takes the dearer-side difference). The “dilution” formula is just compound percentage decrease.

Partnership

What it is. Profit shared in ratio of (Capital × Time).

Formulas.

• Simple (same time): ratio = capitals
• Compound: ratio = C₁T₁ : C₂T₂ : C₃T₃
• Working partner: deduct salary/commission first, then split remainder by capital ratio

Worked. A: 40k×12 = 480, B: 60k×9 = 540 → ratio 8:9.

Remember. Always reduce numbers (don’t carry zeros). For investment changes mid-year, split the year into periods and sum each partner’s C×T.

Algebra & Equations

Linear Equations

What it is. Equation where variable powers are 1.

Methods. Substitution (when one var has coeff 1) or Elimination (match coefficients).

Word translation.

• “sum of two” → x+y
• “twice a” → 2x
• “of” → ×
• “is/was/will be” → =
• 2-digit number with digits a,b → 10a+b; reversed = 10b+a

Three-equation systems. Number of solutions test for a₁x+b₁y=c₁ and a₂x+b₂y=c₂:

• Unique: a₁/a₂ ≠ b₁/b₂
• None: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinite: a₁/a₂ = b₁/b₂ = c₁/c₂

Sum-difference shortcut. Sum 100, diff 20 → numbers are (100/2)±(20/2) = 60 and 40.

Remember. Age problems: “after t years” adds t to every age.

Quadratic Equations

What it is. ax² + bx + c = 0.

Formulas.

• Quadratic: x = (−b ± √(b²−4ac))/2a
• Discriminant D = b² − 4ac. >0: two distinct real; =0: equal; <0: complex
• Sum of roots: −b/a; Product: c/a
• Equation from roots: x² − (sum)x + (product) = 0
• Identity: α² + β² = (α+β)² − 2αβ; α³+β³ = (α+β)³ − 3αβ(α+β)
• 1/α + 1/β = (α+β)/(αβ) = −b/c

Comparison problems (banking exam style). Solve both equations for roots, compare sets.

Remember. D=0 → equal roots = −b/2a. Always check if factorization works before reaching for the formula.

Inequalities

What it is. Like equations but with < > ≤ ≥.

Critical rule. Multiplying/dividing by negative FLIPS the sign.

Wavy curve method (quadratic inequalities). Find roots, plot on number line, alternate signs starting from rightmost region with +, pick regions matching inequality.

Modulus.

• |x| < a ↔ −a < x < a
• |x| > a ↔ x < −a or x > a
• |x − k| < a ↔ k−a < x < k+a

Remember. Never multiply both sides by a variable unless we know its sign. |x| ≥ 0 always — so |x| < negative is impossible.

Arithmetic and Geometric Progressions

What it is. AP adds same number; GP multiplies by same number.

AP formulas.

• aₙ = a + (n−1)d
• Sₙ = (n/2)[2a + (n−1)d] = (n/2)(first + last)
• For 3-term AP, write as (a−d), a, (a+d) — sum simplifies to 3a

GP formulas.

• aₙ = arⁿ⁻¹
• Sₙ = a(rⁿ−1)/(r−1) (r > 1)
• S∞ = a/(1−r) (if |r| < 1)

Useful sums.

• 1+2+…+n = n(n+1)/2
• 1²+2²+…+n² = n(n+1)(2n+1)/6
• 1³+2³+…+n³ = [n(n+1)/2]²
• First n even = n(n+1); first n odd = n²

HP. Reciprocals form AP. HM of a,b = 2ab/(a+b). AM ≥ GM ≥ HM.

Remember. Bouncing ball total distance = first drop × (1+r)/(1−r).

Logarithms

What it is. logₐN = x ↔ aˣ = N. Inverse of exponentiation.

Laws.

• log(AB) = logA + logB
• log(A/B) = logA − logB
• log(Aⁿ) = n logA
• Change of base: logₐB = log B / log a
• Flip: logₐb = 1/log_b a
• logₐa = 1, logₐ1 = 0
• a^(logₐN) = N

Memorize. log 2 ≈ 0.301, log 3 ≈ 0.477, log 7 ≈ 0.845. Derive: log 4=2log2, log 5=1−log2, log 6=log2+log3, log 8=3log2, log 9=2log3.

Number of digits in N = floor(log₁₀N) + 1.

Remember. log(A+B) ≠ logA + logB. There’s no rule for log of a sum. Common log = base 10, ln = base e.

Set Theory and Venn Diagrams

What it is. Counting via overlap formulas.

Formulas.

• 2 sets: n(A∪B) = n(A) + n(B) − n(A∩B)
• 3 sets: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
• Only A = n(A) − n(A∩B); Neither = n(U) − n(A∪B)
• Exactly one of A,B = n(A)+n(B) − 2n(A∩B)

Filling 3-set Venn. Center first → pairwise (subtract center) → onlys (subtract everything overlapping) → outside.

Min/Max bounds.

• Max(A∩B) = min(n(A), n(B))
• Min(A∩B) = max(0, n(A)+n(B) − n(U))

Remember. Always fill the center first; everything else flows from there.

Time, Speed & Work

Time and Work

What it is. Work = Rate × Time. Use LCM method.

LCM method. Total work = LCM of times. Efficiency = Total/Time. Combined = sum efficiencies. Time = Total/Combined.

Formulas.

• A and B together: ab/(a+b)
• MDH: M₁D₁H₁ = M₂D₂H₂ (men × days × hours = constant work)
• Pipes: inlet positive, outlet negative

Worked. A=12, B=18 → LCM=36. A=3/day, B=2/day → 5/day → 36/5 = 7.2 days.

Remember. Combined time is ALWAYS less than the faster individual’s time. If our answer is between the two, we’re wrong.

Time, Speed, and Distance

What it is. D = S × T. Watch units.

Conversions. km/h to m/s × 5/18; reverse × 18/5.

Average speed.

• Equal distances: 2ab/(a+b)
• Equal times: (a+b)/2
• 3 equal distances: 3abc/(ab+bc+ca)

“Late/Early” formula.

• Distance = (S₁ × S₂ × time difference) / (S₁ − S₂)

Relative speed. Same direction: S₁−S₂. Opposite: S₁+S₂.

Remember. Speed-time inversely proportional. Speed up 25% = 5/4 → time down to 4/5. The trap: average speed for round trip is harmonic, not arithmetic.

Trains

What it is. TSD with train length included in distance.

Cases.

• Cross pole/person: distance = train length
• Cross platform: distance = train + platform
• Cross train (opposite): distance = L₁+L₂, speed = S₁+S₂
• Cross train (same dir): distance = L₁+L₂, speed = S₁−S₂
• Man on platform: like pole. Man on another train: distance = passing train’s length, speed = relative

Shortcut. Train crosses pole in t₁, platform L in t₂ → train length = L × t₁/(t₂−t₁).

Remember. Always convert km/h to m/s (×5/18). For two trains, distance is ALWAYS L₁+L₂.

Boats and Streams

What it is. Current helps downstream, fights upstream.

Formulas.

• Downstream = b + s; Upstream = b − s
• b = (down + up)/2; s = (down − up)/2
• Round-trip time = 2Db/(b² − s²)
• b/s = (T_up + T_down)/(T_up − T_down)

Floating object trick. If something falls off the boat, the boat’s relative speed in water is the same both ways. Time to come back = time it has been gone. Object speed = stream speed.

Remember. Stream speed = (downstream − upstream)/2. Use the time-ratio shortcut to skip individual speeds.

Races and Circular Tracks

What it is. Racing vocabulary + circular meeting times.

Linear race.

• “A beats B by x m in D m” → speed ratio = D : (D−x)
• Chain: A beats B in 100 → when B does 90, scale C accordingly
• Head start to make dead heat = the beat distance

Circular track.

• First meeting (opposite): track / (S₁+S₂)
• First meeting (same dir): track / |S₁−S₂|
• Both at start together: LCM(T₁, T₂) where Tᵢ = track/Sᵢ
• Distinct meeting points: (a+b) opposite, |a−b| same dir (speed ratio a:b reduced)

Remember. Same direction = much longer to meet. For 3 runners on circular track, find pairwise meeting times then LCM.

Clocks

What it is. Two hands on a circle = relative speed problem.

Speeds.

• Minute hand: 6°/min; hour hand: 0.5°/min
• Relative: 5.5°/min

Master formula. Angle at H:M = |30H − 5.5M| (subtract from 360° if > 180°).

Counts in 12 hours.

• Coincide: 11 (not 12!)
• 180° apart: 11
• 90° (right angle): 22

Interval between coincidences: 12/11 hours = 65 5/11 minutes.

Faulty clock. Gains/loses → ratio of clock-minutes to real-minutes. Real time = clock time × 60/(60±gain).

Mirror time. 11:60 − mirror time.

Remember. Each hour mark = 30°. 30H gives base position; 5.5M is the relative correction.

Geometry & Mensuration

Lines, Angles, and Triangles

What it is. Angle properties + Pythagoras + similarity.

Angles. Complementary sum to 90°, supplementary to 180°. Parallel lines + transversal: corresponding equal, alt-interior equal, co-interior supplementary.

Triangle.

• Angle sum = 180°
• Exterior angle = sum of two non-adjacent interiors
• Triangle inequality: each side < sum of other two
• Pythagoras: hyp² = base² + height² (right triangle)
• Area = ½ × b × h = √[s(s−a)(s−b)(s−c)] (Heron) = ½ab sin(C)
• Equilateral: area = (√3/4)a², height = (√3/2)a

Pythagorean triplets (memorize): 3-4-5, 5-12-13, 8-15-17, 7-24-25 + multiples.

Similarity. Same shape, different size. AA, SAS, SSS conditions. Area ratio = (side ratio)².

Remember. SSA isn’t valid for congruence. AAA = similarity, not congruence. If c² < a²+b² → acute, c²=a²+b² → right, c²>a²+b² → obtuse.

Circles

What it is. Arcs, chords, tangents and their relationships.

Formulas.

• Circumference = 2πr; Area = πr²
• Arc length = (θ/360) × 2πr
• Sector area = (θ/360) × πr²
• Segment area = sector − triangle
• Annulus = π(R² − r²)

Theorems.

• Perpendicular from center bisects chord → form right triangle
• Tangent ⊥ radius at point of contact
• Two tangents from external point are equal
• Inscribed angle = ½ × central angle (same arc)
• Angle in semicircle = 90°
• Cyclic quadrilateral: opposite angles sum to 180°

Remember. When chord problem given, drop perpendicular from center → use Pythagoras with half-chord. Diameter subtends 90° everywhere.

Quadrilaterals and Polygons

What it is. 4-sided and n-sided figures with diagonal/angle properties.

Areas.

• Parallelogram: base × height
• Rectangle: l × b; diagonal = √(l²+b²)
• Rhombus: ½d₁d₂; side = ½√(d₁²+d₂²)
• Square: a² = ½d²; diagonal = a√2
• Trapezium: ½(a+b)h
• Kite: ½d₁d₂

Diagonal table.

Polygons.

• Sum of interior angles = (n−2) × 180°
• Each interior of regular polygon = (n−2)×180°/n
• Each exterior = 360°/n
• Diagonals = n(n−3)/2
• Sum of exterior angles = 360° (any convex polygon)

Remember. For polygon problems, find exterior angle first (360/n) — simpler than interior. Square is the everything-shape; rhombus and rectangle each add one diagonal property.

Mensuration 2D (Areas and Perimeters)

What it is. Composite shapes, paths, tiling.

Path formulas.

• Outside path width w around l×b: 2w(l+b+2w)
• Inside path: 2w(l+b−2w)
• Around circle: π(R²−r²) = πw(2r+w)

Carpet/tiling. Tiles = floor area / tile area (same units!).

Wire bending. Perimeter constant; among same-perimeter shapes, circle has max area (and square beats other rectangles).

Semicircle perimeter. πr + 2r (curve + diameter), NOT just πr.

Remember. Always convert all units to the same scale before computing tile counts. Add/subtract areas for composites.

Mensuration 3D (Volume and Surface Area)

What it is. Volume, CSA (curved/lateral), TSA (total).

Master table.

Diagonals. Cube: a√3. Cuboid: √(l²+b²+h²). Cone slant: √(r²+h²).

Melting/recasting. Volume conserved → Volume₁ = Volume₂. Sphere → n smaller spheres: n = (R/r)³.

Water flow. Volume/sec = πr²v (pipe). Tank fill time = tank volume / flow rate.

Remember. Cone = ⅓ cylinder of same r,h. Hemisphere TSA includes flat face (3πr², not 2). Always check whether problem gives slant or vertical height.

Coordinate Geometry Basics

What it is. Algebra meets geometry on a plane.

Formulas.

• Distance: √[(x₂−x₁)² + (y₂−y₁)²]
• Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
• Section (m:n internal): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
• Centroid: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
• Slope: (y₂−y₁)/(x₂−x₁)
• Area of triangle: ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
• Parallel: same slope. Perpendicular: m₁m₂ = −1.

Line forms. Slope-intercept y=mx+c; point-slope y−y₁=m(x−x₁); intercept x/a + y/b = 1.

Collinearity. Area = 0 OR pairwise slopes equal.

Remember. Distance formula is just Pythagoras. Look for Pythagorean triplets in differences (saves time).

Counting & Probability

Permutations

What it is. Arrangements where order matters.

Formulas.

• nPr = n!/(n−r)!
• n distinct in line: n!
• With repeats: n!/(p! q! r!…)
• Circular: (n−1)!
• Necklace/bracelet: (n−1)!/2

Memorize factorials. 5!=120, 6!=720, 7!=5040, 8!=40320, 9!=362880, 10!=3628800. (0! = 1.)

Restrictions.

• “Always together” → bundle as one, arrange units, then arrange within
• “Never together” → total − together (or place gaps method)
• For digit problems with even/odd → fix the constrained position FIRST

Remember. MISSISSIPPI = 11!/(4!4!2!) = 34650. Always tackle most-restricted positions first.

Combinations

What it is. Selections where order doesn’t matter.

Formulas.

• nCr = n!/(r!(n−r)!) = nPr/r!
• nCr = nC(n−r) (use smaller!)
• nC0 = nCn = 1; nC1 = n
• nCr + nC(r−1) = (n+1)Cr (Pascal)

Common values. nC2 = n(n−1)/2 (handshakes). nC3 = n(n−1)(n−2)/6.

Diagonals of n-gon = nC2 − n = n(n−3)/2.

Distribution.

• n identical objects, r distinct groups (≥0 each): (n+r−1)C(r−1) (stars and bars)
• n distinct objects, r distinct groups: rⁿ

Restricted committees.

• “X must be in” → (n−1)C(r−1)
• “X must be out” → (n−1)Cr
• “At least k of A and B” → cases by group counts

Remember. “At least 1” = total − none (complement). nCr = nC(n−r) for symmetry — always use smaller one.

Probability

What it is. Favorable / Total. 0 to 1.

Rules.

• P(not E) = 1 − P(E)
• P(A∪B) = P(A) + P(B) − P(A∩B)
• Mutually exclusive: P(A∩B) = 0
• Independent: P(A∩B) = P(A)P(B)
• Conditional: P(A|B) = P(A∩B)/P(B)
• Bayes’: P(A|B) = P(B|A)P(A)/P(B)

Complement is gold. P(at least 1) = 1 − P(none). Always faster.

Remember. Mutually exclusive ≠ independent (they’re actually opposites: ME events are dependent because one happening forces the other to not). For “AND” use multiplication; check independence.

Dice, Coins, and Cards

What it is. Sample spaces and patterns for the three classic objects.

Coins. n coins → 2ⁿ outcomes. Exactly r heads = nCr/2ⁿ. P(at least 1H in n) = 1 − (1/2)ⁿ.

Dice (two dice, 36 outcomes).

P(sum=7) = 1/6 (highest). P(doubles) = 1/6. P(at least one 6 in 2 throws) = 11/36.

Cards (52 deck). 4 suits × 13 ranks. 26 black, 26 red. 12 face cards, 4 aces. P(spade) = 1/4. P(king) = 1/13. P(face) = 3/13.

Remember. “At least” → complement. With/without replacement is critical (denominator changes).

Data Interpretation Basics

What it is. Read tables, bar/line/pie charts, calculate fast.

Pie chart. 1% = 3.6°. 25% = 90°. 50% = 180°. Sector value = (%/100) × Total.

Speed techniques.

• Approximate first; refine only if options are close
• Compare ratios via cross multiplication (no division)
• Use fraction-percent equivalents
• Anchor to nice numbers: 17% = 10% + 5% + 2%

Remember. Always check units (lakhs vs crores). Always use OLD value as base for % change. Don’t recalculate if comparing only — use ratios.

Data Sufficiency

What it is. Decide if given info is enough; don’t solve.

Five answers.

• (A) S1 alone sufficient, S2 not
• (B) S2 alone sufficient, S1 not
• (C) Both together sufficient, neither alone
• (D) Each alone sufficient
• (E) Even both together insufficient

Approach. Test S1 alone first; then S2 alone; combine only if needed.

Common traps.

• Don’t assume positivity, integrality, etc.
• For yes/no questions, sufficient = ALWAYS gives same answer
• x² = 25 gives two values of x (5, −5) — but |x| = 5 uniquely
• 2 unknowns usually need 2 independent equations

Remember. Don’t actually solve — just check determinability. Test edge cases.

Logical Reasoning

Number and Letter Series

What it is. Spot pattern, predict next.

Patterns to check.

1. Constant difference / increasing difference
2. Multiply pattern (×k)
3. Squares/cubes (n², n³, n²±c)
4. Fibonacci (sum of previous two)
5. Alternating operations (+a, ×b, +a, ×b)
6. Two interleaved series (separate odd/even positions)
7. Prime numbers
8. Factorials (1, 2, 6, 24, 120)

Method. Take first differences. If still messy, take second differences. Check ratios for GP.

Letters. Convert to position numbers (A=1, …, Z=26). EJOTY landmarks: E=5, J=10, O=15, T=20, Y=25.

Remember. If random-looking, try separating odd-positioned and even-positioned terms. Wrong-number problems: find pattern, find term that breaks it.

Coding and Decoding

What it is. Crack the cipher.

Types.

• Constant shift (Caesar): each letter moves by k
• Reverse alphabet: A↔Z, B↔Y… (positions sum to 27)
• Position-based shift (1st letter +1, 2nd letter +2, …)
• Number coding: A=1, B=2…
• Word substitution: compare sentences sharing words
• Condition-based (TCS NQT): different rules for vowels/consonants

Method. First check for constant shift (most common). Verify with second pair.

Reverse alphabet check. Original position + coded position = 27 → confirms it.

Remember. ROT13 is its own inverse. For word substitution, find the word common to multiple sentences first.

Blood Relations

What it is. Family tree puzzles.

Approach. ALWAYS draw a family tree. Same generation = same horizontal level. Use + for male, − for female.

Decoding coded phrases. Work inside-out.

• “A’s father’s only son” = A himself
• “A’s mother’s only daughter” = A herself
• “A’s father’s son” = A or A’s brother
• “A’s mother’s husband” = A’s father

Tree notation. -- for marriage, | for parent-child.

Common relations. Father’s brother = paternal uncle; mother’s brother = maternal uncle; uncle/aunt’s children = cousins.

Remember. “Only” is the most important word. “Father’s only son” pins down identity uniquely. In-laws can be ambiguous.

Direction and Distance

What it is. Trace path on paper, find net displacement.

Compass. 8 directions: N, NE, E, SE, S, SW, W, NW. Each adjacent pair = 45°.

Turn rules.

• Right from N → E → S → W → N (clockwise)
• Left from N → W → S → E → N (counter-clockwise)

Method. Draw compass. Mark start. Draw each segment in correct direction with distance label. Cancel opposite movements.

Shortest distance. √(net horizontal² + net vertical²) — Pythagoras.

Shadows.

• Morning sun in East → shadow falls West
• Evening sun in West → shadow falls East

Remember. Always draw the path, never solve in head. Cancel N vs S and E vs W to get net.

Seating Arrangement and Puzzles

What it is. Place people with constraints (linear, circular, two-row, floor).

Approach.

1. Read ALL clues first
2. Place definite clues (“third from left”, “at the right end”)
3. Apply relative clues (“immediate right of B”)
4. Apply negative clues (“not adjacent to D”)
5. Use elimination

Circular gotcha.

• Facing center → left = clockwise
• Facing outside → left = counter-clockwise
• Fix one person to remove rotational symmetry

Linear two-row. Row 1 N-facing and Row 2 S-facing → opposite person sits across; left/right reversed across rows.

Remember. Person with most constraints gets placed first. If stuck with 2 cases, try both.

Syllogisms

What it is. Logical deduction from premises.

Statement types.

• All A are B (A inside B)
• Some A are B (overlap)
• No A are B (disjoint)
• Some A are not B (part of A outside B)

Method. Draw Venn diagrams. Conclusion is definite only if true in ALL valid arrangements. Possibility = true in at least one arrangement.

Complementary pair shortcut. “Some A are B” and “No A are B” → exactly one must be true. Same for “All A are B” and “Some A are not B.” If neither follows individually but they’re complementary, “Either I or II follows.”

Remember. “Some” includes “all.” “All A are B” does NOT mean “All B are A.” Ignore real-world knowledge — only use given premises.

Order and Ranking

What it is. Position from top/bottom, swaps, comparisons.

Master formula. Total = Left + Right − 1 (also Top+Bottom, Start+End).

Position from other end = Total − pos + 1.

People between = |pos₁ − pos₂| − 1.

Swap. A and B exchange → A takes B’s old position, B takes A’s old position.

Height/weight rankings. Convert clues to inequalities, chain them: if A>B and B>C, then A>B>C.

Remember. “−1” matters in the total formula because the person is counted from both sides.

Tips & Exam Strategy

Mental Math and Vedic Tricks

What it is. Speed shortcuts for multiplication and squaring.

Key tricks.

• × 11 (2-digit): put digits’ sum in middle. 36×11 = 3|9|6 = 396. Carry if sum > 9.
• × 5: ÷ 2, × 10
• × 25: ÷ 4, × 100
• × 50: ÷ 2, × 100
• × 9: × 10 − × 1
• × 99: × 100 − itself
• × 101: × 100 + itself (just write number twice for 2-digit)
• × 125: ÷ 8, × 1000

Squaring.

• n5² = n(n+1)|25. 65² = 42|25 = 4225
• Near 50: (50+x)² = (25+x)|x². 53² = 28|09 = 2809
• Near 100: (100+x)² = (100+2x)|x². 97² = 94|09 = 9409
• (a+b)(a−b) = a²−b² for products of equidistant pairs. 97×103 = 100²−3² = 9991

Digit sum verification. Digit sum of operands op-applied = digit sum of answer. Mismatch = wrong.

Remember. Memorize squares 1²–30². Master fraction-percent table. These collectively save 10+ minutes per exam.

Approximation and Option Elimination

What it is. Use options to skip work.

Decision rule. Options far apart → approximate. Options close → compute exactly.

Techniques.

1. Unit digit elimination: check unit digit of answer; eliminate options not matching
2. Parity: even or odd? Eliminate half the options
3. Order of magnitude: is the answer in hundreds or thousands? Eliminate impossible-magnitude options
4. Rounding (opposite directions): errors cancel
5. Back-substitution: for equations, plug each option in
6. Combine techniques: unit digit + magnitude often pins answer in seconds

Worked. 197 × 203 ÷ 401 ≈ 200 × 200 / 400 = 100.

Remember. Look at options BEFORE solving. Sometimes elimination beats arithmetic.

Time Management in Aptitude Tests

What it is. Strategy to maximize marks within time.

Three-pass strategy.

1. Pass 1 (easy): solve only ≤60-second questions; mark rest
2. Pass 2 (medium): 1-2 minute questions
3. Pass 3 (hard/guess): what’s left

The 2-minute rule. Stuck for 2 min with no progress? Mark and move on.

Negative marking math.

• No penalty → always guess
• −0.25 with 4 options → break-even on random; profitable if eliminate ≥1
• −0.33 with 4 options → break-even
• Full negative → guess only if confident

ROI by topic. Highest: percentages, ratio, averages, series. Lowest: permutations, geometry, logarithms (in most exams).

Remember. First-instinct answers are usually right. Don’t change unless sure. Always re-read what’s actually being asked.

Common Traps and Mistakes

What it is. The 10 most common ways to lose marks.

Top traps.

1. +x% then −x% ≠ 0% (it’s actually a x²/100 net loss)
2. % MORE vs % LESS with different bases (A is 25% > B → B is 20% < A)
3. Average speed for equal distances = 2ab/(a+b), NOT (a+b)/2
4. Compare fractions by cross multiplication, not by comparing parts
5. BODMAS — multiplication before addition
6. Misreading the question — re-read after computing
7. Unit mismatch — km/h ↔ m/s, minutes ↔ hours (45 min = 0.75 hr, not 0.45)
8. “Too easy” answer on hard section — often a planted trap
9. (−3)² = 9 but −3² = −9 — parentheses matter
10. Sanity check — is the answer reasonable?

Remember. “After 30% discount price is 910” → original = 910/0.7 = 1300, NOT 910 + 30% of 910. The base is what you don’t have yet.